We present a framework for analyzing the exact dynamics of a class of online
learning algorithms in the high-dimensional scaling limit. Our results are
applied to two concrete examples: online regularized linear regression and
principal component analysis. As the ambient dimension tends to infinity, and
with proper time scaling, we show that the time-varying joint empirical
measures of the target feature vector and its estimates provided by the
algorithms will converge weakly to a deterministic measured-valued process that
can be characterized as the unique solution of a nonlinear PDE. Numerical
solutions of this PDE can be efficiently obtained. These solutions lead to
precise predictions of the performance of the algorithms, as many practical
performance metrics are linear functionals of the joint empirical measures. In
addition to characterizing the dynamic performance of online learning
algorithms, our asymptotic analysis also provides useful insights. In
particular, in the high-dimensional limit, and due to exchangeability, the
original coupled dynamics associated with the algorithms will be asymptotically
"decoupled", with each coordinate independently solving a 1-D effective
minimization problem via stochastic gradient descent. Exploiting this insight
for nonconvex optimization problems may prove an interesting line of future
research.